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निम्न दो (02) प्रश्नों के लिए निम्नलिखित पर विचार कीजिए :
माना कि \(y = \sin^{-1} \left( x - \frac{4x^3}{27} \right)\) है।
\(\frac{dy}{dx}\) किसके बराबर है?
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दिया गया है,
फलन है \( y = 3 \sin^{-1} \left( \frac{x}{3} \right) \), और हमें y का अवकलज, अर्थात \( \frac{dy}{dx} \) ज्ञात करना है।
\( \sin^{-1}(u) \) का सामान्य अवकलज है:
\( \frac{d}{dx} \sin^{-1}(u) = \frac{1}{\sqrt{1 - u^2}} \cdot \frac{du}{dx} \), जहाँ \( u = \frac{x}{3} \)
\( \frac{du}{dx} = \frac{1}{3} \)
y का अवकलज ज्ञात करने के लिए श्रृंखला नियम लागू करने पर:
\( \frac{dy}{dx} = 3 \times \frac{1}{\sqrt{1 - \left( \frac{x}{3} \right)^2}} \times \frac{1}{3} \)
\( \frac{dy}{dx} = \frac{1}{\sqrt{1 - \frac{x^2}{9}}} = \frac{3}{\sqrt{9 - x^2}} \)
इसलिए, सही उत्तर विकल्प 3 है।
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